Problem: Simplify. $i ^ {31}$
Answer: The most important property of the imaginary unit $i$ is that ${i ^ 2} = {-1}$ When this property is plugged into $i ^ 4$ , we get: $i ^ 4 = ({i ^ 2}) ^ 2 = ({-1}) ^ 2 = 1$ So, we can reduce the exponent by multiples of 4 and have the same result. The remainder after dividing 31 by 4 is 3, so $i ^ {31} = i ^ {3}$ As stated above, ${i ^ 2} = {-1}$ $i ^ 3 = ({i ^ 2}) \cdot i = ({-1}) \cdot i = -i$ $i ^ {31} = i ^ {3} = -i$.